In a large braiser or saucepan, sauté the chopped onions gently in olive oil until they are very soft and translucent, taking care that they don't brown. While the onions are braising, season them lightly with salt and pepper, and add a few drops of water from time to time—this will help them soften and prevents browning.

Pour the peas, together with its liquid, into the pot. Let them simmer gently for just a few minutes, to allow the flavors to meld.

While the peas are simmering, whisk the eggs together with the grated cheese and, if using, the parsley, in a mixing bowl. Season generously with salt and pepper.

Turn up the heat a bit and add the egg mixture to the peas. Mix together with a wooden spoon or spatula until the eggs have formed soft curds and remove from the heat. Serve immediate, with some extra cheese sprinkled on top if you like, and perhaps a nice turn or two of freshly ground pepper.